Remember that you are trying to comprehensively explain the theory and applications of 3D with elucidated examples and consistently using subject terminology correctly.
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Mesh Construction:
Mesh construction is a term used for describing different methods of creating something that is called a 'mesh' within computer software. A mesh is a term for a 'polygon', which in terms means intersects between two points on a grid within the computer software. A computer draws with numbers, when we draw a line, what we see as a line, computer sees as a bunch of numbers and coordinates on Y,Z and X axis. Every piece of drawing like that is a polygon, a model will consist of thousands or even millions of polygons. They can be described as the lego blocks of 3D modelling. And thus mesh construction is creating different polygons using different techniques.
Primitive:
Primitive Mesh Construction consists of constructing shapes and figures out of very basic shapes, but by joining them together making more complex shapes. Those shapes includes cubes, spheres, circles, triangles, pyramids, cones, stars etc. With them, any other shape can be created and thus it allows us to create something that can be very complicated with a quite simple shapes and ideas at the start. The shapes can be overlapped so only parts of the can be visible at a time, thus allowing us for better quality modelling.
Box Modelling:
Box modelling is modelling using pure boxes, instead of using something like the primitive shapes, a modeler will just use a simple box for a start. Then he will adjust the box as he wants it, so for example, you can spawn in a box and then model it out to become a foot, or a chest. Make many components, and then join them together into a model. A box can be modeled into anything and everything and therefore it is very versatile and useful for creating shapes out of something very simple at the start.
Extrusion Modelling:
Extrusion Modelling is where an artist creates a 2D model of something in a program like Adobe Illustrator, creating detail on the model however still keeping it in 2D. Then an artists goes into something like a LightWave Modeller and he pastes the model into the program, he then uses a tool within the program called extrusion tool to make the object 3D, i.e have depth. Additional tools can be used within the 3D software to smooth out the edges and give it a less 'blocky' look. An artists can also model additional elements using extrusion, turn them around, flip them and/or drag them out or in. This method is very useful since it requires the artist to put less work in but come out with the same effect at the end, which saves money, time and other resources. Extrusion modelling is therefore widely used among the artists. Main things that are modeled with extrusion are symmetrical or semi-symmetrical things, like weaponry, armor/clothing, human bodies and human faces.
Sketch Modelling:
Sketch Modelling is one of the more difficult and more challenging types of mesh construction to date. It consists of an artist starting out with a pure and bare skeleton of something, whether it would be a building, a human or any other animal. Then, the artists uses a set of specific tools to draw the character/building on the bare bones. It can be very challenging, it requires a wide set of good artistic skills and deep understanding of the software and its tools used. Therefore many artists do not use sketch modelling, however, if done correctly, it can be a very effective and efficient process, and of course it is very good to be able to do this since it will allow you to use other techniques much easily. Sketch modelling can sometimes include the use of primitive or box modelling to help it along.
3D Scanners:
3D mesh construction using scanners is the newest additions to the already extensive ways of mesh construction. What it allows is for an artist to put an object for it to be scanned and then moved onto the software inside a computer. Then an artist can use the tools inside the software to tweak the object the way he wants it. This technique is extremely efficient and time saving and thus some artist call it the future of mesh construction, however there are many restraints to this. Scanning can sometimes turn out to be impossible for certain objects or can be very hard to do to begin with. Of course, the advantages heavily outweigh the disadvantages and thus this method is widely used within certain areas of 3D modelling industry.
Sources:
Images:
http://vimeo.com/38784662
http://www.webreference.com/3d/cararra/3.html
http://sillord.free.fr/wip/tutotopology/textrusion.html
http://blendermama.com/handling-geometry-and-simple-modelling-blender-2-49.html
http://www.cgmeetup.net/home/zbrush-character-creation-process-from-blizzard/
http://www.thegnomonworkshop.com/news/2013/08/3d-scanners-lots-of-zombies-and-world-war-z/